Familias De Conjuntos

Families

There are occasions when the range of a function is deemed to be more important than the function itself. When that is the case, both the terminology and the notation undergo radicalalterations. Suppose, for instance, that x is a function from a set I to a set X (The very choice of letters indicates that something strange is afoot.) An element of the domain I is called an index, I iscalled the index set, the range of the function is called an indexed set, the function itself is called a family, and the value of the function x at an index i, called a term of the family, isdenoted by xi.

(This terminology is not absolutely established, but it is one of the standard choices among related slight variants; in the sequel it and it alone will be used.) An unacceptable butgenerally accepted way of communicating the notation and indicating the emphasis is to speak of a family {z} in X, or of a family {x} of whatever the elements of X may be; when necessary, the index set Iis indicated by some such parenthetical expression as (i∈I).
Thus, for instance, the phrase “a family { A} of subsets of X” is usually understood to refer to a function A, from some set I of indices,into Y(X).
If {A} is a family of subsets of X, the union of the range of the family is called the union of the family {A}, or the union of the sets A; the standard notation for it is

i∈IAi oriAi,

according as it is or is not important to emphasize the index set L It follows immediately from the definition of unions that z E U A if and only if z belongs to A for at least one i. If I = 2,so that the range of the family {A1} is the unordered pair {A0, A1}, then U A = A0 U A.
Observe that there is no loss of generality in considering families of sets instead of arbitrary collections ofsets; every collection of sets is the range of some family. If, indeed, e is a collection of sets, let C itself play the role of the index set, and consider the identity mapping on C in the role of...